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Sojourn time in ℤ+ for the Bernoulli random walk onℤ

Published online by Cambridge University Press:  01 August 2012

Aimé Lachal
Aimé Lachal*
Affiliation:
Institut National des Sciences Appliquées de Lyon, Pôle de Mathématiques/Institut Camille Jordan, Bâtiment Léonard de Vinci, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France. [email protected]; http://maths.insa-lyon.fr.
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Abstract

Let (Sk)k≥1 be the classical Bernoulli random walk on the integer line with jump parametersp ∈ (0,1) andq = 1 − p. The probability distribution of the sojourntime of the walk in the set of non-negative integers up to a fixed time is well-known, butits expression is not simple. By modifying slightly this sojourn time through a particularcounting process of the zeros of the walk as done by Chung & Feller [Proc.Nat. Acad. Sci. USA 35 (1949) 605–608], simpler representationsmay be obtained for its probability distribution. In the aforementioned article, only thesymmetric case (p = q = 1/2) isconsidered. This is the discrete counterpart to the famous Paul Lévy’s arcsine law forBrownian motion.

In the present paper, we write out a representation for this probabilitydistribution in the general case together with others related to the random walk subjectto a possible conditioning. The main tool is the use of generating functions.

Keywords

Random walksojourn timegenerating function
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 324 - 351
Copyright
© EDP Sciences, SMAI, 2012

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